Normal distribution

Chapter 130 of 38 strong

RESTS ONIdentify integrand, variable of integration, lower endpoint and upper endpoint, and distinguish a definite integral's numeric value from an indefinite integral's family of functionsCH 8 Evaluate an exact definite integral as F(upper) - F(lower) using any anti-derivative of the integrand, with the arbitrary constant cancellingCH 8 Interpret the area function A(x) = ∫_a^x f(t) dt as an accumulation, state the fundamental theorem d/dx A(x) = f(x), and connect it to the evaluation rule ∫_a^b f = F(b) − F(a) for any anti-derivative FCH 8 Compute E(X) = ∫ x·f(x) dx and E(g(X)) = ∫ g(x)·f(x) dx over the density's support on exact carriers, distinguishing the mean from the total area and from the most likely valueCH 12 Compute Var(X) = E(X^2) − [E(X)]^2 where both expectations are exact continuous integrals, and compute the standard deviation as the principal square root of the varianceCH 12 Map exact points and endpoint inclusion under translations, reflections in either axis and dilations from either axisCH 2 Signed numbersCH 3 Compute Var(X) = E(X^2) − [E(X)]^2 on exact carriers, and interpret the variance as a spread measure in squared unitsCH 10 Compute SD(X) = √Var(X) as the principal square root of the variance, exactly where the variance is a rational square and at a stated precision otherwiseCH 10 Multiplication and divisionCH 9 Percentage of a quantityCH 14Build and read standard-deviation intervalsFOUNDATIONSComplete and validate a finite probability assignment: every probability is non-negative and the total is exactly one; use stated unequal outcome masses rather than assuming equiprobabilityCH 10Decimal place valueFOUNDATIONSSolve a one-step linear equationCH 7 Solve equations with unknowns on both sides by collecting them onto one sideCH 1 Check a claimed solution or point by substituting into the original equationCH 1 Calculate Pr(a < X < b) = ∫_a^b f(x) dx for a piecewise density by identifying which branch applies on each part of the interval, splitting at branch junctions, and adding exact partial areasCH 12 Interpret the event after the conditioning event is known and restrict the denominator to that event's outcomes or table marginCH 10 Calculate probabilities of specified success/failure patterns across independent trials, including all, only and at-least-one events, using complements where efficientCH 11 Compute a combination coefficient exactly from its factorial ratio, cancelling before multiplication and respecting the selection constraintsCH 11 Map each elementary outcome to a stated finite random-variable value, combine the masses of outcomes with the same value, and complete or read its probability tableCH 10

The normal curve and its parameters

Standard and general normal probabilities

Inverse normal calculations

Normal conditional and mixed applications

Read the mean μ and standard deviation σ from a normal probability density function written in any of the taught equivalent algebraic forms, including the canonical form, a compact exponential coefficient form, and a shifted form with x + c; state the variance as σ^2 and write the distribution in N(μ, σ^2) notation

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Practise this

RESTS ON

Distinguish a continuous random variable from a discrete one, interpret a probability density function as a smooth limiting model of grouped frequency data, and state that Pr(X = x) = 0 at a single value so strict or non-strict interval endpoints do not change an interval probabilityNot startedInterpret f, f(x), image, preimage, rule, domain and co-domain; evaluate a function at a stated exact input and solve a simple equation of the form f(x)=cNot startedMultiply and divide fractionsNot started

UNLOCKS

State and verify the key properties of a normal density: symmetric about μ; μ = median = mode; maximum value 1/(σ√(2π)) attained at x = μ; strictly positive on the real line; tails asymptotic to the horizontal axis; total area exactly one — and distinguish a characteristic that always holds from one that holds only for a particular parameter pairDescribe and identify the graph effect of changing μ or σ: changing μ translates the curve horizontally; changing σ applies a horizontal dilation by σ and a vertical dilation by 1/σ while preserving total area one; a larger σ gives a flatter, wider curve and a smaller σ a taller, narrower curve; and match a described curve to its parameter values